 Hello and welcome to the session. In this session we discussed the following question which says show that 1 upon root 5 minus 2 minus 1 upon root 6 minus root 5 plus 1 upon root 7 minus root 6 minus 1 upon root 8 minus root 7 plus 1 upon 3 minus root 8 is equal to 5. Let's move on to the solution. We need to show 1 upon root 5 minus 2 minus 1 upon root 6 minus root 5 plus 1 upon root 7 minus root 6 minus 1 upon root 8 minus root 7 plus 1 upon 3 minus root 8 is equal to 5. We need to show this. Now first we consider this expression 1 upon root 5 minus 2. We rationalize its denominator and for this we multiply its numerator and the denominator by root 5 plus 2. This would be equal to root 5 plus 2 upon 5 minus 4 that is this is equal to root 5 plus 2 upon 1. Thus we get 1 upon root 5 minus 2 is equal to root 5 plus 2. Now consider this 1 upon root 6 minus root 5. We rationalize its denominator. For this we multiply its numerator and denominator by root 6 plus root 5. So this is equal to root 6 plus root 5 upon 6 minus 5 that is this is equal to root 6 plus root 5 upon 1. Thus we get 1 upon root 6 minus root 5 is equal to root 6 plus root 5. Then we consider 1 upon root 7 minus root 6. Then we rationalize its denominator by multiplying its numerator and denominator by root 7 plus root 6 this is equal to root 7 plus root 6 upon 7 minus 6 that is this is equal to root 7 plus root 6 upon 1 that is we get 1 upon root 7 minus root 6 is equal to root 7 plus root 6. Now the next expression is 1 upon root 8 minus root 7 in this also we would rationalize its denominator by multiplying its numerator and denominator by root 8 plus root 7. So this is equal to root 8 plus root 7 upon 8 minus 7 this is equal to root 8 plus root 7 upon 1 that is we get 1 upon root 8 minus root 7 is equal to root 8 plus root 7. Then we have 1 upon 3 minus root 8 here also we will rationalize its denominator by multiplying its numerator and denominator by 3 plus root 8 and this would be equal to 3 plus root 8 upon 9 minus 8 that is we get this is equal to 3 plus root 8 upon 1 that is we have 1 upon 3 minus root 8 is equal to 3 plus root 8. Now let us consider 1 upon root 5 minus 2 minus 1 upon root 6 minus root 5 plus 1 upon root 7 minus root 6 minus 1 upon root 8 minus root 7 plus 1 upon 3 minus root 8. Now we will substitute the values of all these so this would be equal to now the value for this is root 5 plus 2 minus value for this is root 6 plus root 5 plus value for this which is root 7 plus root 6 minus value for this which is root 8 plus root 7 plus value for this that is 3 plus root 8. So now this is equal to root 5 plus 2 minus root 6 minus root 5 plus root 7 plus root 6 minus root 8 minus root 7 plus 3 plus root 8. Now root 5 and minus root 5 cancels root 7 minus root 7 cancels root 6 minus root 6 cancels root 8 minus root 8 cancels and we are left with 2 plus 3 and this is equal to 5. Thus we get 1 upon root 5 minus 2 minus 1 upon root 6 minus root 5 plus 1 upon root 7 minus root 6 minus 1 upon root 8 minus root 7 plus 1 upon 3 minus root 8 is equal to 5. So hence this completes the session hope you have understood the solution for this question.